Question

The negation of \( p \wedge (q \to r) \) is

• A. \( \sim p \wedge (q \to \sim r) \)
• B. \( p \vee (\sim q \vee r) \)
• C. \( \sim p \vee (q \to \sim r) \)
• D. \( p \to (q \wedge r) \)

Hint:

Use De Morgan's laws to negate compound logical expressions; verify with truth tables if needed.

Answer 1

The Correct Option is B

Step 1: Express \( q \to r \): \( q \to r = \sim q \vee r \).
Thus, \( p \wedge (q \to r) = p \wedge (\sim q \vee r) \).

Step 2: Negation: \( \sim [p \wedge (\sim q \vee r)] \). Using De Morgan's:
\[ \sim (p \wedge (\sim q \vee r)) = \sim p \vee \sim (\sim q \vee r) = \sim p \vee (q \wedge \sim r). \]

Step 3: Check options: None match \( \sim p \vee (q \wedge \sim r) \). Re-evaluate using \( q \to r \):
Negation of \( p \wedge (\sim q \vee r) = \sim p \vee (q \wedge \sim r) \). Test option (b):
\[ p \vee (\sim q \vee r) = \sim [\sim p \wedge \sim (\sim q \vee r)] = \sim [\sim p \wedge (q \wedge \sim r)] = \sim p \vee (q \wedge \sim r). \] This matches the negation.
Answer: \( p \vee (\sim q \vee r) \).